(1) Lj(u) = ldm + (-1) DA u = 0, j = 0,1; k = 1,2,,

Transcription

1 BASIC SETS OF POLYNOMIALS FOR GENERALIZED BELTRAMI AND EULER-POISSON-DARBOUX EQUATIONS AND THEIR ITERATES1 E. P. MILES, JR. AND EUTIQUIO C. YOUNG 1. Introduction. This paper concerns basic sets of polynomial solutions for the class of partial differential equations in m variables, m^2, (1) Lj(u) = ldm + (-1) DA u = 0, j = 0,1; k = 1,2,, where 2 2 Di = d /dxi + (ai/xi)(d/dxi) with a< 3; 0; i = 1,, m. The iterated operators L* are defined by the relations l7 (u) = Lj[Lj(u)], s = 1,, k - 1. When ax= =aro_i = 0 and am>0, Lj(u)=0 is known as the Beltrami or the Euler-Poisson-Darboux (EPD) equation according as.7 = 0 or j= 1. If am = 0 too, then L0(u) =0 and Lx(u) =0 become the Laplace and wave equations, respectively. Basic sets of polynomial solutions for the Laplace and wave equations have been given in a number of papers [l]-[5]. In [6] Miles and Williams obtained basic sets of polynomials for the Beltrami and EPD equations from their result in [3]. In [7] the result of [3] was extended to form basic sets for the iterated Laplace and wave equations. Here we derive basic sets for (1) from the basic sets given in [7]. The Miles and Williams basic set of homogeneous polynomials of degree n for the -fold iterated Laplace equation Aku = 0 (A= XXi d2/dx2) may be represented by [(> -««.)/21. /, 4- \a /2l\ r2;+ (2)Hai...,am= 2w (-1)1 r, )A(x! -xm-x), i=o \ [am/2\ / (2j + am)l where ax,, am are nonnegative integers such that 2^ iat = w and am^2k 1. In particular, when am = 0, H21,...,am-x,o is harmonic, Presented to the Society, January 24, 1967 under the title Basic sets of polynomials for iterated generalized Beltrami and EPD equations; received by the editors September 6, This work was supported by NSF research grant GP

2 982 E. P. MILES, JR. AND E. C YOUNG [December that is, it satisfies Au = 0. We shall prove that if for every index i (1 Hkifkn) such that a,->0 we restrict the a,- to be nonnegative even integers and replace xf' by (2«,->_ 1-3 (25,- 1) 2, \\j) %% Xi i (1 +«*) (2st -1+a,) then (2) gives a basic set for Zj(w)=0 (or L\(u)=0 if the factor ( 1)' is deleted). 2. Basic set for L0(u) =0, s = l, 2,, k. We first observe that any polynomial solution of (1) must be even in the variable xt whenever «i>0, l^i^m. Indeed, suppose <x,>0 and suppose that uix) is a polynomial solution of (1) which contains odd powers of Xj, x=ixi,, xm). Let Pix'), a polynomial of X\,, x;_i, x,-+i,, xm, be the first nonvanishing coefficient of xf+1 when u(x) is arranged in ascending powers of x,. Then the coefficient of x2n+1~2t in L*o(u)=0 would be (2»+l) (2n-2k+3)(2n+af) (2n+aj 2k + 2)Pix'), a nonvanishing function. We assume that at least one of the a/s is not zero. In fact, by changing subscripts if necessary, we can assume ai= =ap = 0, ap+i>0,, am>0, O^p-^m 1. Let oi,, ap, rp+u, rm be a set of nonnegative integers satisfying the condition (4) 2>.- + Z 2fi = N, rm^k- 1, i=\ i=p+\ and let _>( '2' y/i + r A "il,, ap. 'p+...-,tm 2 1 \ 1/ I I i=o \ rm / (5) 2j+2rm j, oi «p 2rp+1 2rm~u X"t_ A (Xi xp xp+i.xm_i ) -. ~ ~ - (2/ + 2r, ) I Denote by T, (p + l^i^m) the operator which replaces xf* by xf3^ (see (3)) in every term of the polynomial (5) and put T=Tp+i Tm.2 Then the operator T applied to (5) replaces each factor x\%t xhm by x^i+l) x[2sm). Moreover, from the fact that Di-Ti=Tid2/dx2i and Dj-Tk=Tk-Dj for jvfe, we see that 2 We are indebted to the referee for suggesting this notation and pointing out that the operator 7", is a special case of a well-known operator treated by Lions, Operateurs de Delsarte et problemes mixtes, Bull. Soc. Math. France 84 (1956); Proposition 2.1, p. 65.

3 1967] BASIC SETS OF POLYNOMIALS 983 J^d2/dXi+ DijT^i Tm 1=1 i=p+l / = T d*/dx* + Tp+X DiTi Tm i=l i=p+l P 2 2 -A 2 2 (6) =r2a /3X,- + ^+1 "Tid /dxi Tm!=1 t=p + l Now let = T^d2/dxl+ T X) d/d*< t=l i=p+l = TA. N N (') Qax,. -,ap,rp+1, -,rm = * ("ax,,ap,rp+x,.,rm). We assert that (7) forms a basic set for L*(u) =0. Lemma 1. io(g^,...«11,r,+1,...,r -1,o)=0. This follows from (6) and the fact that P^.- --,ap,rp+1,---,rm-i,o is harmonic. Lemma 2. LJ(G».... r, , _,,.) = C-28.,ap,r,+1,,r _ o, 0^2* Suppose that y ^ ^_2,' *^»lv:«i..«p.r,+i,,r-_i,y) = y<ii, -.ap.rp+l -,rm_i,0, 0^2j<2(k-l)^N,ai+ +ap + 2rp+i+ + 2rm_i + 2j = N. Then for ax+ +ap+2rp+x+ +2rm_x + 2(j+l) = N, we have Lo iqax,,ap,rp+u.rm-x,j+l) = Lo [L0(T(Pai,...,ap,rp+1,,r _lty+l)) ] = Lo [rap lf...,ap,rp+1.,rm-l,}+l] = Lo[l (Rax,..,ap.rp+1,,rm-X,})J, = Lo(Qax,..,Op,rp+i, -,rm-i,y) _ sf-2-2' - Vax, -,<Jp,rp+1,.,rm_i,0, see (6) of [7], where ax+ +ap+2rp+x+ +2rm-i+2j = N 2. Now we verify that (7) forms a basic set for L\(u)=0. Consider first the case 2(k l)^n. From Lemmas 1 and 2 it follows that all members of (7) satisfy the equation Z,J(w)=0. So we need only to

4 984 E. P. MILES, JR. AND E. C. YOUNG [December show that (7) has the correct number of independent polynomials. For a given integer N^O, it is clear that (7) has as many independent polynomials as there are distinct ways of choosing the set ai, - -, ap, rp+i,, rm which satisfies (4). Let E. "1 _ap (2rp+l) (2r, ^ A.ail...,ap,rp+i,- -,rmxi ' Xv Xp±i Xm, «i+ -r-ap + 2rj,+i+ +2rm = N, O^p^m l, be any homogeneous polynomial of xx,, xm of degree N, N}±2(k 1), which is even in the variables xt, p + l^i^m. Here we have already replaced each x2s< by xf*7 Then every coefficient Aait...,ap,Tp+u...,Tm of u can be represented, apart from constant factor, as 7,,---,vr,+,.---,r ~(«l ' dp Dp+i Dm)u, where d{ = d/dxi, i= 1, 2,, p. If Lk0(u) =0, so that r>': /v *! J2a' J""nr"+l nr"a Dmu = 1 27-«i dp Dp+i Dm \u \ ail aplrp+il - - rml / where ai+ +ap+rp+i+ +rm = k with rm^k I, then every derivative of the form (df dappdp"f) Dr )u can be written in such a way that Dm occurs no more than (k 1) times. Thus, if Lo(u)=0, all coefficients of u are linear combinations of the coefficients A,1,...,tr,tr+1,...,tm.1,M, OSz^k l, where si+ +sp + 2tp+i + +2z = N which coincides with (4). Therefore, for 2(&-l)^ N, the set (7) is correctly numbered and hence forms a basic set for Ll(u)=0. In the case N<2(k l), it is clear that all the members of (7) satisfy Lj(w)=0. In order to prove that (7) has the correct number of polynomials, we examine (4) with rm:s [iv/2] under the following cases. Case 1. N = 2n,n^0. Suppose first that = 2g; then for each s, O^s^n, where s replaces rm, only 2v of the a/s can be chosen as odd integers with 0^z/: [q, n}. Here [q, n] denotes the smaller of the integers q and n. Hence, writing fl, = 2r, if a,- is even and a,= 2r,-f-l if a, is odd (1 ^i^p), we have ai+ +ap = 2ri+ +2rp+2v so that (4) becomes r%+ + rm_i = n s v. Now for each s, O^s^n, the set (7) has i^] /2q\ tm + n-2- s - v\ h'o \2v)\ m-2 ) independent polynomials, since for each of the

5 1967] BASIC SETS OF POLYNOMIALS 985 CI) ways of choosing 2v of the a/s odd the set P2",...,,,_,,«is seen to be generated by the monomials xa/ xavvx2/l+/ x^lf with ai + +aj, = 2ri+ +2^+2^, where xf1 x^zf are the individual terms appearing in (2~l?-il x2)n~*~v. Therefore, for N = 2n and p = 2q, the set (7) consists of n lom /2g\ (m + n 2 s v\ (8) zzzz("){. )»_o v~o \2v/ \ m 2 / independent polynomials homogeneous of degree 2n. Here 0 is interpreted as zero whenever a <b. If p = 2q + l, then (7) will have n [j^n] /2q + n /m 4. n 2 s v\ (9) IE, 9 ),=0»-o \ 2» / \ m 2 / independent polynomials homogeneous of degree 2n. Case 2. N=2n + l, n^o. In this case p^o. This means that no polynomial of odd degree can satisfy Z,J(m)=0 when a,>0, i=l, 2,, m. Again, suppose first that p = 2q; then 2^+1 of the a/s must be chosen odd, Ogfl ^ [q 1, «]. Hence we can write ai+ +ap = 2ri+ +2rp + 2z/+l and again (4) reduces to ri+ +rm_i = n s v. By the same argument as in Case 1, we see that (7) has ao) i"r'(7' \(»+«-*-<-\,=o»=o \2v + 1/ \ m 2 J independent homogeneous polynomials of degree 2n + l. For p = 2q+l, (7) consists of 8=o»=o \2v + 1/ \ m 2 / independent homogeneous polynomials. Noting that for fixed v the summands corresponding to s>n v are all zero, we can carry out the summation with respect to s in the formulas (8) (11). In fact, from (8) for example, we obtain

6 986 E. P. MILES, JR. AND E. C. YOUNG n [^»] /2g\ /m + n_2-s-v\_ t^»l /2<A s_o t>=o \2v/\ m 2 / c_0 \2vJ 1^ /m + n - 2 s v\ _ '«^J /2g\ /j» +» - v 1\,_o \ m 2 / _0 \2z'/V w 1 /' the number of independent polynomials in (7) when N = 2n, n<k l, and p = 2q. Indeed, when N<2(k 1) the basic set elements in case N=2n and p = 2q could be chosen for each v, 0^v^[q, n], as the individual monomials xax xppx2plf x%\m, ax+ +ap = 2rx + +2rp + 2v, where x?xtl x2f? are the individual terms appearing in (Xl-i %i)n~v which has precisely (m + n v 1\ m 1 elements. Members of basic set for the other cases can also be chosen in the same manner thus proving that (7) is correctly numbered. Therefore, for given integers A 3^0, k^l, and the associated sets of nonnegative integers ax,, ap, rp+x,, rm satisfying (4), (2rm^N if N<2(k 1)), the set of polynomials given by (7) is a basic set for L\(u) = Basic set for L\(u)=0. The corresponding basic set of polynomials for L\(u) =0 under the same assumption on the a.-'s as before may be deduced from (7) upon replacement of xm by ixm. We have the same formula as (7) except for the absence of the factor ( 1)'. This fact can of course be established by the same procedure as in 2 using (5) with the factor ( 1)' deleted. References 1. M. H. Protter, Generalized spherical harmonics, Trans. Amer. Math. Soc. 63 (1948), , On a class of harmonic polynomials, Portugal. Math. 10 (1951), E. P. Miles, Jr. and Ernest Williams, A basic set of homogeneous harmonic polynomials in k variables, Proc. Amer. Math. Soc. 6 (1955) J. Horvath, Singular integral operators and spherical harmonics, Trans. Amer. Math. Soc. 82 (1956), , Basic sets of polynomial solutions for partial differential equations, Proc. Amer. Math. Soc. 9 (1958), E. P. Miles, Jr. and Ernest Williams, A basic set of polynomial solutions for the Euler-Poisson-Darboux and Beltrami equations, Amer. Math. Monthly 63 (1956), , Basic sets of polynomials for the iterated Laplace and wave equations, Duke Math. J. 26 (1959), 35^0. Florida State University I